1. Field of the Invention
The present invention relates to a determination method of determining a light intensity distribution (effective light source) to be formed on the pupil plane of an illumination optical system, an exposure method, and a storage medium.
2. Description of the Related Art
An exposure apparatus is employed to fabricate a semiconductor device using a photolithography technique. The exposure apparatus projects and transfers the pattern of a mask (reticle) onto a substrate (for example, a wafer) by a projection optical system. To keep up with the recent advances in micropatterning of semiconductor devices, the exposure apparatus requires a technique of attaining a high resolution.
Because an exposure apparatus cannot always ensure an ideal amount of exposure on the substrate and an ideal focus position, it may transfer a pattern different from that having a desired shape (mask pattern shape) onto a substrate. The amount of exposure deviates from an ideal state due to factors such as instability of a light source and nonuniformity of the illuminance distribution in an illumination region. Also, the focus position deviates from an ideal state due to factors such as instability of the holding position of the substrate and unevenness of the substrate. A model defined by the ranges of amounts of exposure and focus positions, within which a desired pattern can be transferred onto the substrate, is called a process window, and the exposure apparatus requires a technique of attaining a wide process window.
Oblique-incidence illumination, for example, is known as a technique for attaining both a high resolution and a wide process window. In the oblique-incidence illumination, a mask is obliquely irradiated with exposure light using an annular effective light source (the light intensity distribution on the pupil plane of an illumination optical system) or an effective light source with a shape having a plurality of (for example, two or four) poles. The annular effective light source is defined by two degrees of freedom (parameters): the annular zone radius and the annular zone width. Thus, the following technique has been proposed. Pattern images for effective light sources defined by those two degrees of freedom are obtained by simulation while changing them to various values, and an annular zone radius and an annular zone width are selected based on these pattern images, thereby determining an optimum effective light source.
Also, in recent years, T. Matsuyama, et. al., “A Study of Source & Mask Optimization for ArF Scanners”, Proc. of SPIE, USA, SPIE, 2009, Vol. 7,274, p. 727,408 (literature 1), proposes a technique which increases the number of degrees of freedom which define the effective light source. In the technique described in literature 1, the pupil plane of an illumination optical system is divided into a plurality of regions in a grid pattern, and light intensities are individually set for the respective divided regions. However, assuming, for example, that the pupil plane of the illumination optical system is divided into 63×63 regions, a thousand or more degrees of freedom are determined. From the viewpoint of the computation time, it is not realistic to obtain pattern images for respective combinations of degrees of freedom defined within such a wide optimization space to determine an optimum effective light source. Although Japanese Patent No. 3342631 proposes a heuristic optimization technique of adjusting the initial value and iterating computation to obtain an optimum solution, this technique may not only require a long computation time but also result in a local solution.
On the other hand, Japanese Patent No. 4378266 and Japanese Patent Laid-Open No. 2002-261004 propose techniques which use mathematical programming in effective light source optimization with such large degrees of freedom. The mathematical programming mathematically guarantees its solution to be optimum, and can shorten the computation time.
The technique described in Japanese Patent No. 4378266 is designed to apply approximation to a maximization problem for the process window to transform this problem into one type of mathematical programming, that is, a linear programming problem to be solved, thereby obtaining a solution. The process window is generally the product of the range of amounts of exposure and that of focus positions, within which the width of a pattern image falls within a tolerance. However, in the technique described in Japanese Patent No. 4378266, the position coordinates of the two side edges of a line pattern image and the ranges of these position coordinates are defined, instead of defining the width (line width) of the line pattern image. Therefore, in the technique described in Japanese Patent No. 4378266, the effective light source is optimized by evaluating the intensities of the line pattern image at the positions at which its two edges are to be positioned (that is, by indirectly evaluating the width of the line pattern image), instead of directly evaluating the width of the line pattern image.
Also, the technique described in Japanese Patent Laid-Open No. 2002-261004 is designed to optimize the effective light source and the mask pattern so that a pattern image having a desired shape is formed. In the technique described in Japanese Patent Laid-Open No. 2002-261004, a two-dimensional pattern to be formed is determined in advance, and a plurality of position coordinates (image points) are set on the two-dimensional pattern. A light state or a dark state is then defined for each of the plurality of position coordinates, and the effective light source and the mask pattern are optimized. Therefore, the technique described in Japanese Patent Laid-Open No. 2002-261004 lacks the concept of the width of the pattern image, like that described in Japanese Patent No. 4378266.
In the techniques described in Japanese Patent No. 4378266 and Japanese Patent Laid-Open No. 2002-261004, the effective light source is optimized in consideration of the light/dark state of the pattern image at each position coordinate in place of the width of the pattern image. Unfortunately, such optimization is inappropriate for actual conditions because the width of the pattern image is of prime importance in evaluating the pattern image. Nevertheless, if the pattern image has bilateral or horizontal symmetry (or vertical symmetry) and the projection optical system has none of factors which act to shift the pattern image, such as distortion and coma, the techniques described in Japanese Patent No. 4378266 and Japanese Patent No. 2002-261004 may pose no problems. This is because if the pattern image is symmetric, the position coordinates of the edges of the pattern image and the width of the pattern image have a one-to-one correspondence between them. However, such a case is rare in practice, and the pattern image shifts (a pattern shift occurs) horizontally (or vertically) with respect to the mask in many cases. When a pattern shift occurs, neither of the techniques described in Japanese Patent No. 4378266 and Japanese Patent Laid-Open No. 2002-261004 can be used to determine an optimum effective light source.